Question: Divide the following complex numbers: $\dfrac{4 e^{5\pi i / 4}}{2 e^{4\pi i / 3}}$ (The dividend is plotted in blue and the divisor in plotted in green. Your current answer will be plotted orange.)
Solution: Dividing complex numbers in polar forms can be done by dividing the radii and subtracting the angles. The first number ( $4 e^{5\pi i / 4}$ ) has angle $\frac{5}{4}\pi$ and radius 4. The second number ( $2 e^{4\pi i / 3}$ ) has angle $\frac{4}{3}\pi$ and radius 2. The radius of the result will be $\frac{4}{2}$ , which is 2. The difference of the angles is $\frac{5}{4}\pi - \frac{4}{3}\pi = -\frac{1}{12}\pi$ The angle $-\frac{1}{12}\pi$ is negative. A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $-\frac{1}{12}\pi + 2 \pi = \frac{23}{12}\pi$ The radius of the result is $2$ and the angle of the result is $\frac{23}{12}\pi$.